Thermochemistry: Enthalpy and Standard Enthalpy Changes
Enthalpy, Heat Capacity, and their Relationship
Enthalpy (H) is defined as the sum of internal energy and the PV term:
H = U + PV- Enthalpy change is a convenient way to quantify heat transfer at constant pressure.
- At constant pressure, the heat exchanged equals the enthalpy change:
The lecture emphasizes measuring enthalpy to relate to heat transfer rather than relying only on internal energy.
Heat capacity in terms of enthalpy:
- At constant pressure, the heat capacity is defined as
CP = \left(\frac{\partial H}{\partial T}\right)P - At constant volume, the heat capacity relates to internal energy:
CV = \left(\frac{\partial U}{\partial T}\right)V - If Cp and Cv are constant, the temperature dependencies can be integrated to relate changes in H and U to temperature changes.
- At constant pressure, the heat capacity is defined as
Relationship between Cp and Cv (for ideal gases):
- A key result (to be shown later in the course) is
CP - CV = nR
where n is the number of moles and R is the gas constant. - For a molar basis, C{P,m} - C{V,m} = R.
- For gases, Cp is typically larger than Cv because part of the energy goes into PV work during expansion.
- For liquids, Cp is roughly equal to Cv because the work term is small (almost no expansion work).
- A key result (to be shown later in the course) is
Practical implication for gases vs liquids:
- Gases: Cp > Cv due to non-negligible expansion work.
- Liquids/solids: Work is negligible, so Cp ≈ Cv;
thus the enthalpy change at constant pressure is not dramatically different from the internal energy change.
Integration under constant capacity assumptions (when Cp/Cv are constants):
- Change in internal energy:
\Delta U = \int C_V \, dT - Change in enthalpy:
\Delta H = \int C_P \, dT - If the heat capacities are constant and a process occurs at constant pressure:
\Delta H \approx C_P \Delta T - If the process is at constant volume:
\Delta U \approx C_V \Delta T
- Change in internal energy:
Transition from thermodynamic quantities to industrial relevance:
- Understanding heat transfer and energy requirements is crucial for processes like hydrogen production (DOE context discussed later).
- In industry, heat needs to be supplied or removed to control temperature and avoid runaway reactions.
Quick primer on why enthalpy is useful: state functions and path independence
- Enthalpy is a state function, like internal energy.
- Consequence: the enthalpy change between an initial state A and a final state B is independent of the path taken from A to B.
- If you go directly from A to B, or via intermediate states A -> X -> B, the same (\Delta H_{A\to B}) results.
- This underpins Hess’s Law (to be covered in detail later): the enthalpy change of a reaction is the sum of enthalpy changes for individual steps.
Enthalpy change of physical transitions (examples):
- Vaporization (liquid to gas):
- Standard enthalpy change of vaporization:
\Delta H_{\text{vap}}^\circ - Example value for water at 1 atm and around 100°C: \Delta H_{\text{vap}}^\circ \approx 40.6\ \text{kJ mol}^{-1}
- Note: occurs at the boiling point of water (100°C) under standard pressure; the process requires input of energy (endothermic).
- Sublimation (solid to gas): typically a positive enthalpy change, involves phase transitions through an intermediate state.
- Mixing/combining substances to form solutions: enthalpy changes can be positive or negative depending on interactions; in ideal mixing, some enthalpy changes can be negligible.
Thermochemical reactions: marrying chemistry with heat transfer
- A thermochemical equation is a chemical equation augmented with a heat term, usually the standard enthalpy change (at standard states):
- Standard enthalpy change for a reaction is denoted as
\Delta_r H^\circ - The standard state convention and standard conditions:
- Standard states refer to pure substances in their standard forms (e.g., pure liquid water, pure gaseous O2) at 1 bar.
- The temperature associated with standard enthalpy changes is conventionally 298 K, but the notation does not explicitly include temperature.
- The standard enthalpy change is reported at 298 K (commonly denoted as 25°C in many textbooks) unless stated otherwise.
- The standard reaction enthalpy is derived from the enthalpies of the products and reactants, accounting for stoichiometric coefficients:
- General form for a reaction aA + bB -> cC + dD:
\Deltar H^\circ = \sumi \nui Hi^\circ(\text{products}) - \sumj \nuj Hj^\circ(\text{reactants}) where ( \nui ) are the stoichiometric coefficients of products, and ( \nu_j ) are those of reactants. - A more formal and common way to compute these values uses standard enthalpies of formation ((\Deltaf H^\circ)): \Deltar H^\circ = \sumi \nui \Deltaf Hi^\circ(\text{products}) - \sumj \nuj \Deltaf Hj^\circ(\text{reactants})
- Example given in the lecture for a combustion process (thermochemical equation):
- Methane combustion (per mole of CH$4$): \ce{CH4 + 2 O2 -> CO2 + 2 H2O} \Deltar H^\circ \approx -890\ \text{kJ mol}^{-1}
- This illustrates that the enthalpy change is tied to the initial and final state enthalpies, weighted by stoichiometric coefficients.
- The sign convention:
- Negative (\Delta_r H^\circ) indicates an exothermic reaction (heat released to surroundings).
- Positive (\Delta_r H^\circ) indicates an endothermic reaction (heat absorbed from surroundings).
Standard states and practical context highlighted in the lecture: Hydrogen production in the U.S. energy system
- The Department of Energy context notes that hydrogen production is a key part of the economy, including fertilizer production that relies on ammonia synthesis.
- The main industrial route for hydrogen is methane steam reforming (natural gas reforming):
- Overall idea: methane reacts with water (steam) under heat to produce hydrogen and carbon-containing byproducts.
- The basic reaction steps described:
- First step (steam reforming):
\ce{CH4 + H2O -> CO + 3 H2} - Second step (water-gas shift):
\ce{CO + H2O -> CO2 + H2}
- First step (steam reforming):